Abstract

In this talk, we shall introduce a series of efficient and accurate neural network methods for solving elliptic interface problems and applications. There are several novel features in the present network; namely, (i) it is a single network architecture regardless of the number of existed interfaces; (ii) the solution and derivative jump discontinuities are accurately captured by simply augmenting an extra interface feature input, (iii) the network can be completely shallow (one-hidden-layer) if the solutions are not complex, (iv) it is completely mesh-free so the problems in irregular domains with irregular interfaces can be handled easily. Numerical results show better efficiency in terms of the number of trainable parameters used, and better accuracy than the traditional finite difference method such as the immersed interface method.